| L(s) = 1 | + 2.28·2-s + 3.21·4-s + 3.72·7-s + 2.77·8-s − 4.52·11-s − 4.87·13-s + 8.50·14-s − 0.0987·16-s − 0.0939·17-s + 7.35·19-s − 10.3·22-s + 1.41·23-s − 11.1·26-s + 11.9·28-s + 6.50·29-s + 5.98·31-s − 5.76·32-s − 0.214·34-s + 37-s + 16.7·38-s − 0.274·41-s + 7.48·43-s − 14.5·44-s + 3.23·46-s + 6.61·47-s + 6.87·49-s − 15.6·52-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.60·4-s + 1.40·7-s + 0.979·8-s − 1.36·11-s − 1.35·13-s + 2.27·14-s − 0.0246·16-s − 0.0227·17-s + 1.68·19-s − 2.20·22-s + 0.295·23-s − 2.18·26-s + 2.26·28-s + 1.20·29-s + 1.07·31-s − 1.01·32-s − 0.0367·34-s + 0.164·37-s + 2.72·38-s − 0.0428·41-s + 1.14·43-s − 2.19·44-s + 0.477·46-s + 0.965·47-s + 0.982·49-s − 2.17·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.070272365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.070272365\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 7 | \( 1 - 3.72T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 + 0.0939T + 17T^{2} \) |
| 19 | \( 1 - 7.35T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 6.50T + 29T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 41 | \( 1 + 0.274T + 41T^{2} \) |
| 43 | \( 1 - 7.48T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 0.507T + 59T^{2} \) |
| 61 | \( 1 + 2.96T + 61T^{2} \) |
| 67 | \( 1 - 7.08T + 67T^{2} \) |
| 71 | \( 1 - 7.51T + 71T^{2} \) |
| 73 | \( 1 - 1.90T + 73T^{2} \) |
| 79 | \( 1 - 0.537T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 + 0.259T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.48070950826671216573281071466, −7.23585154277714624468279549187, −6.10272863249288321168484774976, −5.39133751086114226530274041465, −4.85778138615049270664331230978, −4.69366862114144129120905366178, −3.59509004620616255007496171338, −2.57557650161692755656424643024, −2.39729908108133528335879683868, −0.957323204639655661763257247142,
0.957323204639655661763257247142, 2.39729908108133528335879683868, 2.57557650161692755656424643024, 3.59509004620616255007496171338, 4.69366862114144129120905366178, 4.85778138615049270664331230978, 5.39133751086114226530274041465, 6.10272863249288321168484774976, 7.23585154277714624468279549187, 7.48070950826671216573281071466
